Abstract
Large Gauge Transformations (LGT) are gauge transformations that do not vanish at infinity. Instead, they asymptotically approach arbitrary functions on the conformal sphere at infinity. Recently, it was argued that the LGT should be treated as an infinite set of global symmetries which are spontaneously broken by the vacuum. It was established that in QED, the Ward identities of their induced symmetries are equivalent to the Soft Photon Theorem. In this paper we study the implications of LGT on the S-matrix between physical asymptotic states in massive QED. In appose to the naively free scattering states, physical asymptotic states incorporate the long range electric field between asymptotic charged particles and were already constructed in 1970 by Kulish and Faddeev. We find that the LGT charge is independent of the particles' momenta and may be associated to the vacuum. The soft theorem's manifestation as a Ward identity turns out to be an outcome of not working with the physical asymptotic states.
Highlights
In their construction, the dressed incoming and outgoing states diagonalize the interacting asymptotic Hamiltonian
Large Gauge Transformations (LGT) are gauge transformations that do not vanish at infinity
It was argued that the LGT should be treated as an infinite set of global symmetries which are spontaneously broken by the vacuum
Summary
We review the essential background material and use it to establish our notations. We will mostly work in the retarded coordinates parametrization of flat four dimensional space-time These are convenient when discussing the isometries of future null infinity, where future asymptotic photons are localized (see figure 1). Throughout this paper, we use bold face letters to represent three-vectors In these coordinates the flat spacetime metric, ds2 = −dt2 + dx21 + dx22 + dx, takes the form ds2 = −du2 − 2du dr + 2r2γzzdz dz , where. When working in Lorentz gauge, ∇μAμ = 0, LGT are parametrized by a function satisfying the appropriate wave equation Under such transformations, Aμ → Aμ + ∂μλ , where ∂μ∂μλ = 0. This enables LGT that approach an arbitrary function on the conformal sphere, lim λ (u, r, z, z) = ε (z, z) + O r−1. In this paper we treat future null infinity exclusively, all of our results can be derived for past null infinity in the same exact manner.
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